How to find the value range of a function and how to find the value range of a function

A: Your question is very big and difficult to answer in detail. But there are two big principles:

1).First, the definition domain of the function is found, and the discussion of the value range of the function must be carried out within the definition domain of the function, which is often ignored by beginners.

2).What method to use must be based on the form and nature of the function, there is no fixed method. Broadly speaking, there are probably the following.

Some methods: Derivative method: If you have learned derivatives, then you can use derivatives to find the extreme and maximum values of the function in a defined domain or specified interval;

Inverse function method: The definition domain of the inverse function is the value range of the direct function, and the definition domain is much easier to find than the value range;

Fundamental inequality method: If you can solve it with a fundamental inequality, it is a very pleasant thing;

Limit method: When some defined domains are r, or when the function has infinite discontinuity points, you can consider using the limit evaluation range;

Functional property method: such as quadratic functions, trigonometric functions, logarithmic functions, exponential functions, etc., all have some special properties that can be used;

Others: Functional splitting, decomposition, collocation, transformation, etc. are all methods that can be considered;

It is required that the value range should consider the definition domain of the analytic formula and the increase or decrease of the function, and the maximum and minimum values of the function can be obtained under this condition.

First, find the function relation; 2: It's OK to find a defined domain and substitute it into the technical calculation.

1.Direct method: Starting from the range of independent variables, the value range is deduced.

2.Observation method: For some simple functions, the value range of the function can be directly obtained according to the defined domain and correspondence.

3.Matching method: (or the minimum value method) to find the maximum value and the minimum value, then the value range will come out.

Example: y=x 2+2x+3x [-1,2].

Recipe first, get y=(x+1) 2+1

ymin=(-1+1)^2+2=2

ymax=(2+1)^2+2=11

4.Splitting method: For fractional functions of the form y=cx+d, ax+b, you can split them into a constant and a fraction, and then it is easy to observe the value range of the function.

5.Single-minded tonality: y≠caThe monotonicity of some functions is easy to see. Or first prove the monotonicity of the function, and then use the monotonicity of the function to find the value range of the function.

6.The combination of numbers and shapes, the type of the question is that the analytic formula of the function has an obvious geometric significance, such as the distance formula of two points, the slope of the straight line, etc., if this kind of problem uses the combination of numbers and shapes, it will often be simpler, clear at a glance, and pleasing to the eye.

7.Discriminant method: Using the idea of equations, the equation has a real root evaluation range according to the quadratic judgment equation.

8.Commutation method: Applies to functions with root numbers.

Example: y=x- (1-2x).

Let (1-2x)=t(t 0).

x=(1-t^2)/2

y=(1-t^2)/2-t

t^2/2-t+1/2

1/2(t+1)^2+1

t≥0，∴y∈（-1/2）

9: Image method, directly draw a picture to see the value range.

This is a piecewise function that allows you to see the range at a glance after you draw a graph.

10: Inverse function method. The defined domain of the inverted function is the domain of the original function.

Example: y=(3x-1) (3x-2).

First, find the inverse function y=(2x-1) (3x-3).

The domain is clearly defined as x≠1

So the range of the original function is y≠1

The methods for finding the range of values of a function are:

1. Matching method Formulate the function formula into a vertex format, and then obtain the value range of the function according to the definition domain of the function.

2. Constant separation This is generally for functions in the form of fractions, and the functions on the numerator are arranged in the same form as the denominator as much as possible, and the constant separation is carried out to obtain the value range.

3. Inverse method For the form of y=a certain x, you can use the inverse method, which is expressed as x=a certain y, and at this time, you can see the limit range of y, which is the value range of the original formula.

Fourth, the commutation method For a certain part of the function, which is more complex or unfamiliar, the commutation method can be used to transform the function into a familiar form and solve it.

5. Monotonicity You can find the monotonicity of the function first (pay attention to find the defined domain first), and find the value range of the function on the defined domain according to the monotonicity.

6. Fundamental Inequalities According to the basic inequalities we have learned, we can convert functions into forms that can be used to evaluate the value domain.

7. Combination of numbers and shapes According to the formula given by the function, the graph of the function can be drawn, and the corresponding points can be found on the graph to find the value range.

8. Derivative Method Find the derivative of the function, observe the definition domain of the function, compare the endpoint value with the extreme value, and find the maximum and minimum values, and then you can obtain the value range.

In the classical definition of a function, the range of values that change due to the change of the dependent variable is called the range of values of the function, and in the modern definition of function, it refers to the set of all the images corresponding to all the elements in the definition domain under a certain corresponding law. f:a b, the range is a subset of set b.

For example, f(x)=x, then the range of f(x) is the range of the function f(x).

There are many methods, 1. Some functions have a value range, such as sine function and cosine function, and the value range is [-1,,1], 2. Use the monotonicity of the function to find the maximum value, such as a parabola, the monotonicity is different on the left and right of the axis of symmetry, so the vertex is its maximum value.

3. It is a common method to use the derivative to find the maximum value of the function.

4. The value range of the function is obtained by combining the properties of the image function through the number form.